Math 6610 • Analysis of Numerical Methods I

{ODEs} Consider the initial value problem for a system of ODEs
\(y'=Ay\) where \(A\) has eigenvalues that are widely spread in the left half plane.
Which of the following schemes would be the best choice for solving this
problem? Justify your answer in terms of stability, accuracy, and efficiency.
\[
\begin{aligned}
  (1)\quad y^{n+1} &= y^n+kAy^n,\\
  (2)\quad y^{n+1} &= y^n+kAy^{n+1},\\
  (3)\quad y^{n+1} &= y^n+\frac{k}{2}A\left(y^n+y^{n+1}\right).
\end{aligned}
\]

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